No dollar is a bastard, goes the old saying. That is unless the source of the money is Stand For Children, a nonprofit group that spins off super PACs to support candidates who want to limit teacher seniority, extend the school day, expand early childhood education, and lift the arbitrary cap on charter schools.

This week, Stand for Children extended its hand — one capable of holding as much as $500,000 — to Boston mayoral candidate John Connolly. So what does Connolly do? He sniffs it. Smiles approvingly, at first. And then bites the hand that is trying to feed him. “I did not ask for any money from outside groups,’’ intoned Connolly.

Why would Connolly reject an education reform group that aligns perfectly with his education policies? It’s certainly not disgust on the part of the candidate for independent expenditure committees. That’s the formal term for the super PACs that are allowed to advocate for candidates by raising unlimited sums of money from corporations, unions, individuals, and organizations provided they don’t coordinate their activities with the campaigns. Up until this week, Connolly didn’t seem to have a problem with Democrats for Education Reform — another super PAC — that was quietly canvassing on his behalf.

Quietly, it turns out, is the key word. Stand for Children can be as subtle as a garbage can falling down a flight of stairs. We’re talking about a group whose national cofounder, Jonah Edelman, once reviewed his own performance in the field as “blunt,’’ “presumptuous,’’ and “ungenerous.’’ In recent speeches, Connolly has been trying to de-escalate tensions with the Boston Teachers Union as a way to attract progressive parents to his campaign. Along comes Stand for Children promising a “full-frontal assault’’ on the candidate’s behalf. Connolly is afraid to risk the association.

For all its clumsiness, Stand for Children knows what needs to be done to elevate the quality of urban classrooms. In 2010, the Massachusetts chapter was instrumental in getting the Legislature to lift the moratorium on charter schools in urban school districts. More recently, it negotiated successfully with statewide teachers’ unions to revamp teacher evaluations and place less emphasis on seniority during layoffs. And it’s among the few groups in town that can match the fund-raising strength of the 110,000-member Massachusetts Teachers Association.

Instead of distancing himself from Stand For Children, Connolly should have pushed aside the bogus “Boston Pledge’’ stuck in his face by flailing candidate Rob Consalvo. By signing the pledge, candidates are supposedly doing their part to keep “agenda-driven outside special interest groups’’ out of the Boston mayoral race. In the case of Stand for Children, the outsider tag is nonsense. Regardless of anyone’s opinion of the group, the Massachusetts chapter has been operating for 10 years and offering support to local candidates, including town meeting members and state senators. Local director Jason Williams said that “virtually all of our money that is spent in Massachusetts is raised in Massachusetts.’’

The Boston Teachers Union, meanwhile, is mobilizing its troops to hold on to one of the shortest work days in urban education. Mayoral candidates Martin Walsh and Charlotte Golar Richie are ready to rake in outside contributions from labor unions and women’s groups, respectively. Walsh and Golar Richie won’t be undermining their chances by signing any pledges. And their supporters won’t be sitting on the sidelines. Nor should they. Boston isn’t going to slip into the sea because a bunch of union activists, hard-driving women, or corporate bigwigs decide to pool their resources and spend it on television ads, canvassing, or other forms of political speech on behalf of a favored candidate.

For all its clumsiness, Stand for Children knows what needs to be done to elevate the quality of urban classrooms.

As for the “Boston Pledge,’’ Consalvo could have done a real service by focusing on full and timely disclosures of the contributors to super PACs, instead of trying to drive them out of town. The disclosure rules for these groups aren’t nearly as opaque as critics charge. But they aren’t models of transparency, either. Typical end-of-the-year filings, for example, are next to useless for informing the choices of voters.

The fight to improve Boston’s schools will be long, hard, noisy, and expensive. It’s not for the squeamish or those who would disarm unilaterally. Until this week, Connolly looked like he was more than up to the task. Suddenly, a shadow of doubt appears.

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